The rewrite relation of the following TRS is considered.
active#(first(0,X)) |
→ |
mark#(nil) |
(22) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(23) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(24) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(25) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(26) |
active#(from(X)) |
→ |
cons#(X,from(s(X))) |
(27) |
active#(from(X)) |
→ |
from#(s(X)) |
(28) |
active#(from(X)) |
→ |
s#(X) |
(29) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(30) |
mark#(first(X1,X2)) |
→ |
first#(mark(X1),mark(X2)) |
(31) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(32) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(33) |
mark#(0) |
→ |
active#(0) |
(34) |
mark#(nil) |
→ |
active#(nil) |
(35) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(36) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(37) |
mark#(s(X)) |
→ |
mark#(X) |
(38) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(39) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(40) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(41) |
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(42) |
mark#(from(X)) |
→ |
from#(mark(X)) |
(43) |
mark#(from(X)) |
→ |
mark#(X) |
(44) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(45) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(46) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(47) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(48) |
s#(mark(X)) |
→ |
s#(X) |
(49) |
s#(active(X)) |
→ |
s#(X) |
(50) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(51) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(52) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(53) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(54) |
from#(mark(X)) |
→ |
from#(X) |
(55) |
from#(active(X)) |
→ |
from#(X) |
(56) |
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(23) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(30) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(26) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(32) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(33) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(36) |
mark#(s(X)) |
→ |
mark#(X) |
(38) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(39) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(41) |
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(42) |
mark#(from(X)) |
→ |
mark#(X) |
(44) |
1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 2 and the following precedence and weight functions
prec(first) |
= |
1 |
|
weight(first) |
= |
2 |
|
|
|
prec(0) |
= |
0 |
|
weight(0) |
= |
2 |
|
|
|
prec(nil) |
= |
2 |
|
weight(nil) |
= |
4 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(first) |
= |
[1,2] |
π(s) |
= |
1 |
π(cons) |
= |
1 |
π(mark#) |
= |
1 |
π(mark) |
= |
1 |
π(from) |
= |
1 |
π(active) |
= |
1 |
π(0) |
= |
[] |
π(nil) |
= |
[] |
the
pairs
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(23) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(32) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(33) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
prec(0) |
= |
1 |
|
weight(0) |
= |
2 |
|
|
|
prec(nil) |
= |
2 |
|
weight(nil) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(first) |
= |
2 |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(from) |
= |
1 |
π(cons) |
= |
1 |
π(s) |
= |
[1] |
π(active) |
= |
1 |
π(0) |
= |
[] |
π(nil) |
= |
[] |
the
pair
mark#(s(X)) |
→ |
mark#(X) |
(38) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(from) |
= |
2 |
|
weight(from) |
= |
1 |
|
|
|
prec(s) |
= |
0 |
|
weight(s) |
= |
2 |
|
|
|
prec(0) |
= |
1 |
|
weight(0) |
= |
2 |
|
|
|
prec(nil) |
= |
3 |
|
weight(nil) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(first) |
= |
2 |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(from) |
= |
[1] |
π(cons) |
= |
1 |
π(s) |
= |
[] |
π(active) |
= |
1 |
π(0) |
= |
[] |
π(nil) |
= |
[] |
the
pairs
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(26) |
mark#(from(X)) |
→ |
mark#(X) |
(44) |
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(41) |
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(41) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(46) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(45) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(47) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(48) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[first#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(46) |
|
1 |
≥ |
1 |
2 |
> |
2 |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(45) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(47) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(48) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(50) |
s#(mark(X)) |
→ |
s#(X) |
(49) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(50) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(49) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(52) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(51) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(53) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(54) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(52) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(51) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(53) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(54) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
from#(active(X)) |
→ |
from#(X) |
(56) |
from#(mark(X)) |
→ |
from#(X) |
(55) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[from#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
from#(active(X)) |
→ |
from#(X) |
(56) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.